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The problem was the well known one about a finite group G, |G| = pq where p<q primes. The object was to use Sylow Theorems to show that there is a unique (up to isomorphism) non-abelian group G if, and only if, p|q-1. Of course it is simple to show the case where p does not divide q-1, and my problem came from proving the other case. My question about that case is: Can one prove that there is such a unique non-abelian group without using any theory of automorphisms? Specifically is there a clever group action that can deliver the desired result?

The reason I ask is that our course has not mentioned automorphisms one bit. The text we are using is Ash

*Abstract Algebra: The Basic Graduate Year*, though most of our problems are pulled from Herstein's

*Topics in Algebra*. I have managed to read through the Herstein and pull out the tools I need for the problem, but could I have done the problem with the theory that has so far been provided? So far in a nutshell we have had the isomorphism theorems, the definition of group action, the orbit stabilizer theorem, and the class equation.

Thank you for your time. Cheers.